↑

Properties

Label 2-1456-13.4-c1-0-7
Degree $2$
Conductor $1456$
Sign $-0.603 - 0.797i$
Analytic cond. $11.6262$
Root an. cond. $3.40972$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.583 + 1.01i)3-s + 1.81i·5-s + (0.866 − 0.5i)7-s + (0.817 + 1.41i)9-s + (2.40 + 1.38i)11-s + (−3.58 + 0.402i)13-s + (−1.83 − 1.05i)15-s + (1.37 + 2.37i)17-s + (5.08 − 2.93i)19-s + 1.16i·21-s + (−3.49 + 6.06i)23-s + 1.70·25-s − 5.41·27-s + (1.75 − 3.04i)29-s + 2.06i·31-s + ⋯
L(s)  = 1  + (−0.337 + 0.583i)3-s + 0.811i·5-s + (0.327 − 0.188i)7-s + (0.272 + 0.472i)9-s + (0.725 + 0.418i)11-s + (−0.993 + 0.111i)13-s + (−0.473 − 0.273i)15-s + (0.332 + 0.576i)17-s + (1.16 − 0.673i)19-s + 0.254i·21-s + (−0.729 + 1.26i)23-s + 0.341·25-s − 1.04·27-s + (0.326 − 0.565i)29-s + 0.371i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.603 - 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $-0.603 - 0.797i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ -0.603 - 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.434038596\)
\(L(\frac12)\) \(\approx\) \(1.434038596\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (3.58 - 0.402i)T \)
good3 \( 1 + (0.583 - 1.01i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 1.81iT - 5T^{2} \)
11 \( 1 + (-2.40 - 1.38i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.37 - 2.37i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.08 + 2.93i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.49 - 6.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.75 + 3.04i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.06iT - 31T^{2} \)
37 \( 1 + (-1.50 - 0.871i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.51 - 3.18i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.55 + 7.88i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.65iT - 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + (2.66 - 1.53i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.540 + 0.936i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.34 + 2.50i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.35 - 1.35i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 7.67iT - 73T^{2} \)
79 \( 1 - 15.7T + 79T^{2} \)
83 \( 1 - 7.97iT - 83T^{2} \)
89 \( 1 + (13.9 + 8.03i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (12.3 - 7.11i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.772617401110101443358624708384, −9.394299989597220053613677774544, −7.963716745497652636023243985485, −7.40048868427250929958481671760, −6.63468509897627132173932224425, −5.54829926802458060696777193493, −4.76054028246131734165471731574, −3.93388685397036649023238814769, −2.84177903607948442873735910632, −1.58185820442094019714218984465, 0.63791387113811251964903519194, 1.61910522459514403230913263526, 3.04336744400959260168669515076, 4.26875694049631233179646828879, 5.09998689010116835671919447127, 5.95534192122521564416959067831, 6.78645507997884418980626643268, 7.63693238231898496221771789159, 8.358788752759880699650133655618, 9.338784002934475569993819208725

Graph of the $Z$-function along the critical line